Optimal. Leaf size=147 \[ -\frac {1}{36} b d^3 n x^6-\frac {b e^3 n x^{3 (2+r)}}{9 (2+r)^2}-\frac {3 b d e^2 n x^{2 (3+r)}}{4 (3+r)^2}-\frac {3 b d^2 e n x^{6+r}}{(6+r)^2}+\frac {1}{6} \left (d^3 x^6+\frac {2 e^3 x^{3 (2+r)}}{2+r}+\frac {9 d e^2 x^{2 (3+r)}}{3+r}+\frac {18 d^2 e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right ) \]
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Rubi [A]
time = 0.25, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {276, 2371, 12,
14} \begin {gather*} \frac {1}{6} \left (d^3 x^6+\frac {18 d^2 e x^{r+6}}{r+6}+\frac {9 d e^2 x^{2 (r+3)}}{r+3}+\frac {2 e^3 x^{3 (r+2)}}{r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{36} b d^3 n x^6-\frac {3 b d^2 e n x^{r+6}}{(r+6)^2}-\frac {3 b d e^2 n x^{2 (r+3)}}{4 (r+3)^2}-\frac {b e^3 n x^{3 (r+2)}}{9 (r+2)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 276
Rule 2371
Rubi steps
\begin {align*} \int x^5 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{6} \left (d^3 x^6+\frac {2 e^3 x^{3 (2+r)}}{2+r}+\frac {9 d e^2 x^{2 (3+r)}}{3+r}+\frac {18 d^2 e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {1}{6} x^5 \left (d^3+\frac {18 d^2 e x^r}{6+r}+\frac {9 d e^2 x^{2 r}}{3+r}+\frac {2 e^3 x^{3 r}}{2+r}\right ) \, dx\\ &=\frac {1}{6} \left (d^3 x^6+\frac {2 e^3 x^{3 (2+r)}}{2+r}+\frac {9 d e^2 x^{2 (3+r)}}{3+r}+\frac {18 d^2 e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} (b n) \int x^5 \left (d^3+\frac {18 d^2 e x^r}{6+r}+\frac {9 d e^2 x^{2 r}}{3+r}+\frac {2 e^3 x^{3 r}}{2+r}\right ) \, dx\\ &=\frac {1}{6} \left (d^3 x^6+\frac {2 e^3 x^{3 (2+r)}}{2+r}+\frac {9 d e^2 x^{2 (3+r)}}{3+r}+\frac {18 d^2 e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} (b n) \int \left (d^3 x^5+\frac {18 d^2 e x^{5+r}}{6+r}+\frac {9 d e^2 x^{5+2 r}}{3+r}+\frac {2 e^3 x^{5+3 r}}{2+r}\right ) \, dx\\ &=-\frac {1}{36} b d^3 n x^6-\frac {b e^3 n x^{3 (2+r)}}{9 (2+r)^2}-\frac {3 b d e^2 n x^{2 (3+r)}}{4 (3+r)^2}-\frac {3 b d^2 e n x^{6+r}}{(6+r)^2}+\frac {1}{6} \left (d^3 x^6+\frac {2 e^3 x^{3 (2+r)}}{2+r}+\frac {9 d e^2 x^{2 (3+r)}}{3+r}+\frac {18 d^2 e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 156, normalized size = 1.06 \begin {gather*} \frac {1}{36} x^6 \left (6 b d^3 n \log (x)+d^3 \left (6 a-b n-6 b n \log (x)+6 b \log \left (c x^n\right )\right )+\frac {4 e^3 x^{3 r} \left (-b n+3 a (2+r)+3 b (2+r) \log \left (c x^n\right )\right )}{(2+r)^2}+\frac {27 d e^2 x^{2 r} \left (-b n+2 a (3+r)+2 b (3+r) \log \left (c x^n\right )\right )}{(3+r)^2}+\frac {108 d^2 e x^r \left (-b n+a (6+r)+b (6+r) \log \left (c x^n\right )\right )}{(6+r)^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.27, size = 4021, normalized size = 27.35
method | result | size |
risch | \(\text {Expression too large to display}\) | \(4021\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 218, normalized size = 1.48 \begin {gather*} -\frac {1}{36} \, b d^{3} n x^{6} + \frac {1}{6} \, b d^{3} x^{6} \log \left (c x^{n}\right ) + \frac {1}{6} \, a d^{3} x^{6} + \frac {b e^{3} x^{3 \, r + 6} \log \left (c x^{n}\right )}{3 \, {\left (r + 2\right )}} + \frac {3 \, b d e^{2} x^{2 \, r + 6} \log \left (c x^{n}\right )}{2 \, {\left (r + 3\right )}} + \frac {3 \, b d^{2} e x^{r + 6} \log \left (c x^{n}\right )}{r + 6} - \frac {b e^{3} n x^{3 \, r + 6}}{9 \, {\left (r + 2\right )}^{2}} + \frac {a e^{3} x^{3 \, r + 6}}{3 \, {\left (r + 2\right )}} - \frac {3 \, b d e^{2} n x^{2 \, r + 6}}{4 \, {\left (r + 3\right )}^{2}} + \frac {3 \, a d e^{2} x^{2 \, r + 6}}{2 \, {\left (r + 3\right )}} - \frac {3 \, b d^{2} e n x^{r + 6}}{{\left (r + 6\right )}^{2}} + \frac {3 \, a d^{2} e x^{r + 6}}{r + 6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 868 vs.
\(2 (137) = 274\).
time = 0.38, size = 868, normalized size = 5.90 \begin {gather*} \frac {6 \, {\left (b d^{3} r^{6} + 22 \, b d^{3} r^{5} + 193 \, b d^{3} r^{4} + 864 \, b d^{3} r^{3} + 2088 \, b d^{3} r^{2} + 2592 \, b d^{3} r + 1296 \, b d^{3}\right )} x^{6} \log \left (c\right ) + 6 \, {\left (b d^{3} n r^{6} + 22 \, b d^{3} n r^{5} + 193 \, b d^{3} n r^{4} + 864 \, b d^{3} n r^{3} + 2088 \, b d^{3} n r^{2} + 2592 \, b d^{3} n r + 1296 \, b d^{3} n\right )} x^{6} \log \left (x\right ) - {\left ({\left (b d^{3} n - 6 \, a d^{3}\right )} r^{6} + 22 \, {\left (b d^{3} n - 6 \, a d^{3}\right )} r^{5} + 1296 \, b d^{3} n + 193 \, {\left (b d^{3} n - 6 \, a d^{3}\right )} r^{4} - 7776 \, a d^{3} + 864 \, {\left (b d^{3} n - 6 \, a d^{3}\right )} r^{3} + 2088 \, {\left (b d^{3} n - 6 \, a d^{3}\right )} r^{2} + 2592 \, {\left (b d^{3} n - 6 \, a d^{3}\right )} r\right )} x^{6} + 4 \, {\left (3 \, {\left (b r^{5} + 20 \, b r^{4} + 153 \, b r^{3} + 558 \, b r^{2} + 972 \, b r + 648 \, b\right )} x^{6} e^{3} \log \left (c\right ) + 3 \, {\left (b n r^{5} + 20 \, b n r^{4} + 153 \, b n r^{3} + 558 \, b n r^{2} + 972 \, b n r + 648 \, b n\right )} x^{6} e^{3} \log \left (x\right ) + {\left (3 \, a r^{5} - {\left (b n - 60 \, a\right )} r^{4} - 9 \, {\left (2 \, b n - 51 \, a\right )} r^{3} - 9 \, {\left (13 \, b n - 186 \, a\right )} r^{2} - 324 \, b n - 324 \, {\left (b n - 9 \, a\right )} r + 1944 \, a\right )} x^{6} e^{3}\right )} x^{3 \, r} + 27 \, {\left (2 \, {\left (b d r^{5} + 19 \, b d r^{4} + 136 \, b d r^{3} + 456 \, b d r^{2} + 720 \, b d r + 432 \, b d\right )} x^{6} e^{2} \log \left (c\right ) + 2 \, {\left (b d n r^{5} + 19 \, b d n r^{4} + 136 \, b d n r^{3} + 456 \, b d n r^{2} + 720 \, b d n r + 432 \, b d n\right )} x^{6} e^{2} \log \left (x\right ) + {\left (2 \, a d r^{5} - {\left (b d n - 38 \, a d\right )} r^{4} - 16 \, {\left (b d n - 17 \, a d\right )} r^{3} - 144 \, b d n - 8 \, {\left (11 \, b d n - 114 \, a d\right )} r^{2} + 864 \, a d - 96 \, {\left (2 \, b d n - 15 \, a d\right )} r\right )} x^{6} e^{2}\right )} x^{2 \, r} + 108 \, {\left ({\left (b d^{2} r^{5} + 16 \, b d^{2} r^{4} + 97 \, b d^{2} r^{3} + 282 \, b d^{2} r^{2} + 396 \, b d^{2} r + 216 \, b d^{2}\right )} x^{6} e \log \left (c\right ) + {\left (b d^{2} n r^{5} + 16 \, b d^{2} n r^{4} + 97 \, b d^{2} n r^{3} + 282 \, b d^{2} n r^{2} + 396 \, b d^{2} n r + 216 \, b d^{2} n\right )} x^{6} e \log \left (x\right ) + {\left (a d^{2} r^{5} - {\left (b d^{2} n - 16 \, a d^{2}\right )} r^{4} - 36 \, b d^{2} n - {\left (10 \, b d^{2} n - 97 \, a d^{2}\right )} r^{3} + 216 \, a d^{2} - {\left (37 \, b d^{2} n - 282 \, a d^{2}\right )} r^{2} - 12 \, {\left (5 \, b d^{2} n - 33 \, a d^{2}\right )} r\right )} x^{6} e\right )} x^{r}}{36 \, {\left (r^{6} + 22 \, r^{5} + 193 \, r^{4} + 864 \, r^{3} + 2088 \, r^{2} + 2592 \, r + 1296\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1586 vs.
\(2 (137) = 274\).
time = 1.70, size = 1586, normalized size = 10.79 \begin {gather*} \frac {6 \, b d^{3} n r^{6} x^{6} \log \left (x\right ) + 108 \, b d^{2} n r^{5} x^{6} x^{r} e \log \left (x\right ) - b d^{3} n r^{6} x^{6} + 6 \, b d^{3} r^{6} x^{6} \log \left (c\right ) + 108 \, b d^{2} r^{5} x^{6} x^{r} e \log \left (c\right ) + 132 \, b d^{3} n r^{5} x^{6} \log \left (x\right ) + 54 \, b d n r^{5} x^{6} x^{2 \, r} e^{2} \log \left (x\right ) + 1728 \, b d^{2} n r^{4} x^{6} x^{r} e \log \left (x\right ) - 22 \, b d^{3} n r^{5} x^{6} + 6 \, a d^{3} r^{6} x^{6} - 108 \, b d^{2} n r^{4} x^{6} x^{r} e + 108 \, a d^{2} r^{5} x^{6} x^{r} e + 132 \, b d^{3} r^{5} x^{6} \log \left (c\right ) + 54 \, b d r^{5} x^{6} x^{2 \, r} e^{2} \log \left (c\right ) + 1728 \, b d^{2} r^{4} x^{6} x^{r} e \log \left (c\right ) + 1158 \, b d^{3} n r^{4} x^{6} \log \left (x\right ) + 12 \, b n r^{5} x^{6} x^{3 \, r} e^{3} \log \left (x\right ) + 1026 \, b d n r^{4} x^{6} x^{2 \, r} e^{2} \log \left (x\right ) + 10476 \, b d^{2} n r^{3} x^{6} x^{r} e \log \left (x\right ) - 193 \, b d^{3} n r^{4} x^{6} + 132 \, a d^{3} r^{5} x^{6} - 27 \, b d n r^{4} x^{6} x^{2 \, r} e^{2} + 54 \, a d r^{5} x^{6} x^{2 \, r} e^{2} - 1080 \, b d^{2} n r^{3} x^{6} x^{r} e + 1728 \, a d^{2} r^{4} x^{6} x^{r} e + 1158 \, b d^{3} r^{4} x^{6} \log \left (c\right ) + 12 \, b r^{5} x^{6} x^{3 \, r} e^{3} \log \left (c\right ) + 1026 \, b d r^{4} x^{6} x^{2 \, r} e^{2} \log \left (c\right ) + 10476 \, b d^{2} r^{3} x^{6} x^{r} e \log \left (c\right ) + 5184 \, b d^{3} n r^{3} x^{6} \log \left (x\right ) + 240 \, b n r^{4} x^{6} x^{3 \, r} e^{3} \log \left (x\right ) + 7344 \, b d n r^{3} x^{6} x^{2 \, r} e^{2} \log \left (x\right ) + 30456 \, b d^{2} n r^{2} x^{6} x^{r} e \log \left (x\right ) - 864 \, b d^{3} n r^{3} x^{6} + 1158 \, a d^{3} r^{4} x^{6} - 4 \, b n r^{4} x^{6} x^{3 \, r} e^{3} + 12 \, a r^{5} x^{6} x^{3 \, r} e^{3} - 432 \, b d n r^{3} x^{6} x^{2 \, r} e^{2} + 1026 \, a d r^{4} x^{6} x^{2 \, r} e^{2} - 3996 \, b d^{2} n r^{2} x^{6} x^{r} e + 10476 \, a d^{2} r^{3} x^{6} x^{r} e + 5184 \, b d^{3} r^{3} x^{6} \log \left (c\right ) + 240 \, b r^{4} x^{6} x^{3 \, r} e^{3} \log \left (c\right ) + 7344 \, b d r^{3} x^{6} x^{2 \, r} e^{2} \log \left (c\right ) + 30456 \, b d^{2} r^{2} x^{6} x^{r} e \log \left (c\right ) + 12528 \, b d^{3} n r^{2} x^{6} \log \left (x\right ) + 1836 \, b n r^{3} x^{6} x^{3 \, r} e^{3} \log \left (x\right ) + 24624 \, b d n r^{2} x^{6} x^{2 \, r} e^{2} \log \left (x\right ) + 42768 \, b d^{2} n r x^{6} x^{r} e \log \left (x\right ) - 2088 \, b d^{3} n r^{2} x^{6} + 5184 \, a d^{3} r^{3} x^{6} - 72 \, b n r^{3} x^{6} x^{3 \, r} e^{3} + 240 \, a r^{4} x^{6} x^{3 \, r} e^{3} - 2376 \, b d n r^{2} x^{6} x^{2 \, r} e^{2} + 7344 \, a d r^{3} x^{6} x^{2 \, r} e^{2} - 6480 \, b d^{2} n r x^{6} x^{r} e + 30456 \, a d^{2} r^{2} x^{6} x^{r} e + 12528 \, b d^{3} r^{2} x^{6} \log \left (c\right ) + 1836 \, b r^{3} x^{6} x^{3 \, r} e^{3} \log \left (c\right ) + 24624 \, b d r^{2} x^{6} x^{2 \, r} e^{2} \log \left (c\right ) + 42768 \, b d^{2} r x^{6} x^{r} e \log \left (c\right ) + 15552 \, b d^{3} n r x^{6} \log \left (x\right ) + 6696 \, b n r^{2} x^{6} x^{3 \, r} e^{3} \log \left (x\right ) + 38880 \, b d n r x^{6} x^{2 \, r} e^{2} \log \left (x\right ) + 23328 \, b d^{2} n x^{6} x^{r} e \log \left (x\right ) - 2592 \, b d^{3} n r x^{6} + 12528 \, a d^{3} r^{2} x^{6} - 468 \, b n r^{2} x^{6} x^{3 \, r} e^{3} + 1836 \, a r^{3} x^{6} x^{3 \, r} e^{3} - 5184 \, b d n r x^{6} x^{2 \, r} e^{2} + 24624 \, a d r^{2} x^{6} x^{2 \, r} e^{2} - 3888 \, b d^{2} n x^{6} x^{r} e + 42768 \, a d^{2} r x^{6} x^{r} e + 15552 \, b d^{3} r x^{6} \log \left (c\right ) + 6696 \, b r^{2} x^{6} x^{3 \, r} e^{3} \log \left (c\right ) + 38880 \, b d r x^{6} x^{2 \, r} e^{2} \log \left (c\right ) + 23328 \, b d^{2} x^{6} x^{r} e \log \left (c\right ) + 7776 \, b d^{3} n x^{6} \log \left (x\right ) + 11664 \, b n r x^{6} x^{3 \, r} e^{3} \log \left (x\right ) + 23328 \, b d n x^{6} x^{2 \, r} e^{2} \log \left (x\right ) - 1296 \, b d^{3} n x^{6} + 15552 \, a d^{3} r x^{6} - 1296 \, b n r x^{6} x^{3 \, r} e^{3} + 6696 \, a r^{2} x^{6} x^{3 \, r} e^{3} - 3888 \, b d n x^{6} x^{2 \, r} e^{2} + 38880 \, a d r x^{6} x^{2 \, r} e^{2} + 23328 \, a d^{2} x^{6} x^{r} e + 7776 \, b d^{3} x^{6} \log \left (c\right ) + 11664 \, b r x^{6} x^{3 \, r} e^{3} \log \left (c\right ) + 23328 \, b d x^{6} x^{2 \, r} e^{2} \log \left (c\right ) + 7776 \, b n x^{6} x^{3 \, r} e^{3} \log \left (x\right ) + 7776 \, a d^{3} x^{6} - 1296 \, b n x^{6} x^{3 \, r} e^{3} + 11664 \, a r x^{6} x^{3 \, r} e^{3} + 23328 \, a d x^{6} x^{2 \, r} e^{2} + 7776 \, b x^{6} x^{3 \, r} e^{3} \log \left (c\right ) + 7776 \, a x^{6} x^{3 \, r} e^{3}}{36 \, {\left (r^{6} + 22 \, r^{5} + 193 \, r^{4} + 864 \, r^{3} + 2088 \, r^{2} + 2592 \, r + 1296\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^5\,{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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